Exploring the Chain Rule | Finding the Derivative of a Function with a Variable Raised to a Function

d/dx a^u

To find the derivative of a function involving a variable raised to a function, we can use the chain rule

To find the derivative of a function involving a variable raised to a function, we can use the chain rule. Let’s consider the function f(x) = a^u, where a is a constant and u is a function of x.

Using the chain rule, the derivative of f(x) with respect to x, denoted as df/dx or d/dx(a^u), is calculated as follows:

– First, we differentiate the base function a^u with respect to u, and then multiply it by the derivative of the exponent u with respect to x.

The derivative of a^u with respect to u can be found using the natural logarithm. Specifically, the derivative of a constant raised to a variable with respect to that variable is given by ln(a) * a^u.

The derivative of u with respect to x can be denoted as du/dx.

Putting it all together, we have:

d/dx(a^u) = ln(a) * a^u * du/dx

So, the derivative of a^u with respect to x is ln(a) times a^u times the derivative of u with respect to x.

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